Guarantee valuation in Notional Defined Contribution pension systems

Transcription

1 Guarantee valuation in Notional Defined Contribution pension systems Jennifer Alonso García (joint work with Pierre Devolder) Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA) Université Catholique de Louvain Voie du Roman Pays, 20 bte B-1348-Louvain-la-Neuve, BELGIUM Phone: +32 (0) , Fax +32 (0) /06/2015

2 Outline 1 Introduction Aim Pension Systems in a nutshell Notional Defined Contribution Issue 2 The Setting The Methodology The Results 3 Graphics Interpretation 4

3 Aim Pension Systems in a nutshell Notional Defined Contribution Issue Aim of this talk The aim of this presentation is to: describe the notional defined contribution (NDC) pension scheme with its advantages and shortocomings; develop a pricing framework for guarantees in NDC first pillar pension systems; and compare it to the insurance and financial markets framework.

4 Aim Pension Systems in a nutshell Notional Defined Contribution Issue Overview of pension systems Basic financing techniques Pay as you go (PAYG): current contributors pay current pensioners (Unfunded schemes) Funding: contributions are accumulated in a fund which earns a market return (Funded schemes) Benefit formulae Defined Benefit: Pension is calculated according to a fixed formula which usually depends on the members salary and the number of contribution years. Defined Contribution: Pension is dependent on the amount of money contributed each year and their return. Notional Accounts: Mix of PAYG and Defined Contribution!

5 Aim Pension Systems in a nutshell Notional Defined Contribution Issue Notional Defined Contribution What s NDC? A pension scheme managed by the State, with Pay-as-you-go (PAYG) as financing technique: current contributors pay current pensioners. The pension formula depends on the amount contributed and the return which equal to the notional rate. At retirement age: Accumulated capital Annuity which takes into account: Life expectancy of the individual The indexation of pensions and technical interest rate We assume that the notional rate is equal to the rate of increase of the total contributions.

6 Aim Pension Systems in a nutshell Notional Defined Contribution Issue Notional Defined Contribution (C td) Mathematically, the pension for someone retiring at age x r at time t can be stated as: where: P(x r, t) = x r 1 NDC(xr, t) a(x r, t) NDC(x r, t) = C(x, t x r + x) }{{} x=x 0 yearly contributions t i=t x r +x ( 1 + nr }{{} i ) notional rate (1) and a(x r, t) is the whole time annuity at time t for age x r. It is crucial to have an appropriate capital in order to have an adequate pension!

7 Aim Pension Systems in a nutshell Notional Defined Contribution Issue Advantages of NDC It s more or less actuarially fair (takes into account life expectancy and contributions) Portability of pension rights between jobs, occupations and sectors is permitted. It promises to deal with the effects of population ageing more or less automatically. Arbitrariness in benefit indexation rules and adjustment factors is avoided.

8 Aim Pension Systems in a nutshell Notional Defined Contribution Issue Challenge faced Disadvantages of a pure NDC system (w/out guarantees): The return risk is fully borne by the participants. The amount of pension at retirement is unknown! The notional rate depends on demography s and salary s rate of increase: it can be close to 0% or negative! Why should we consider guarantees? We offer a minimum value for each contribution at retirement. However, which is the price for the guarantee provider?...and how can we price the option on an asset which is not traded in the market? Answer: pricing in incomplete markets techniques!

9 Aim Pension Systems in a nutshell Notional Defined Contribution Issue Utility indifferent pricing Definition The indifference price on an option is defined as the function price which makes these two situations equivalent: hold a financial portfolio (which maximizes capital and hedges the option) and hold (resp. sell) the written option at time t and receive (resp. pay) the payoff at maturity; only hold the financial portfolio. We are in an incomplete market, so the price will depend on the utility function used and the risk aversion γ. In our case, we use the exponential utility: U(x) = e γx, γ > 0 (2)

10 The Setting The Methodology The Results The pension system The working population pays a contribution rate π (0, 1) of their income to the pension system; each contribution earns a rate of return (notional rate) based on the total contribution base at time t. The total contribution base is then: Y (t) = πp(t)l(t), where: P(t) represents the working population and L(t) is the mean salary earned by the workers. Their evolution is driven by a Geometric Brownian Motion ( lognormally distributed)

11 The Setting The Methodology The Results The pension system (C td) For the pension system provider or state, the guarantee scheme entails: holding the contribution base which represents the returns of contribution, a loss of K Y (T ) Y (t) Mathematically, it becomes the put option: with if the contribution base performed badly g(y (T )) = ( K Y (T ) ) + (3) Y (t) dy (t) = d(πp(t)l(t)) = (g + α P + ρ L,P σ L σ P ) Y (t) dt + σ L Y (t)dw L (t) + σ r Y (t)dw P (t) }{{}}{{}}{{} drift salary risk demographic risk (4)

12 The Setting The Methodology The Results Incomplete markets Reminder: The salary and population risks are not traded in the market the market is incomplete! We can t use the classical models! (e.g. Black & Scholes) However, the risks linked to the contribution are correlated to those which are traded! We can then price the underlying non-hedgeable risks, by using the traded risks as a proxy.

13 The Setting The Methodology The Results Financial portfolio We create a self-financing portfolio with the assets available in the market: with: dx (s) X (s) ds0(s) = θ0(s) S + θ P(s) dp(s, T ) 0(s) P(s, T ) + θ S(s) ds(s) S(s) = (r(s) + θ S (s)λ S σ S θ P (s)qσ(s, T ))dt + θ S (s)λ S dw S (s) θ P (s)qσ(s, T )dw r (s) (5) X t = x 0 t s T θ 0(s) proportion invested in the bank account S 0(s); θ S (s) proportion invested in the risky asset S(s); θ P (s) proportion invested in the zero-coupon bond P(s, T ). Constraint: θ 0(s) + θ S (s) + θ P (s) = 1

14 The Setting The Methodology The Results The price Proposition ( ) The price of a European option G = g Y (T ) under the exponential utility in Y (t) a market like the one presented is given by p(x, y, r, t) = P Q (t, T ) δ [ ) ]) (E γ log Q T Y (t) = y (6) e γ δ g ( Y (T ) Y (t) where: δ constant which depends on the level of correlation between the pension risks and the traded risks; γ is the risk aversion; dq T dp is the forward measure;

15 The Setting The Methodology The Results Particular cases: Independent & Complete markets If the notional index is totally independent we find a zero-utility insurance premium: p(x, y, r, t) = P Q (t, T ) 1 ( [ ( ) ]) γ log E P e γg Y (T ) Y (t) Y (t) = y (7) If the notional index is traded we find a complete markets price: [ ( ) ] Y (T ) p(x, y, r, t) = P Q (t, T )E Q T g Y (t) = y Y (t) = Black & Scholes Formula! (8)

16 Graphics Interpretation Comparison between the intermediate price, the insurance and B&S Price Figure : The value of a put option guaranteeing i G = 4% (first row), i G = 6% (second row): un-correlated exponential price (continuous line), imperfect correlation exponential price (discontinuous line) and perfect correlation B&S case (pointed line) Source: the authors. γ = 1 γ = 3 γ = 5

17 Graphics Interpretation Interpretation Link between pure insurance, intermediate price and Black & Scholes: For some risk aversions and guarantees we can have lower prices than Black & Scholes. I m ready to pay more for a better hedge. For the other graphics: Price when risk aversion. I m ready to pay a higher price to insure me against losses when I m more afraid of them. Price with guaranteed rate. I m ready to pay a higher price linked to higher guarantees.

18 Graphics Interpretation Cost of the guarantee: An example Assume the following economy: Salary (Gross - Annual) ,00 Salary rate of increase 1% Contribution rate 16,86% Life expectancy at 65 20,02 Then the Monthly Minimum Pensions (MMP) and the Total Guarantee Cost (TGC) for different minimum returns are: 0% 1% 2% 3% 4% TGC (% of capital) 2,26% 4,14% 6,69% 9,27% 11,53% MMP - State 1.505, , , , ,86 MMP - Contributors 1.471, , , , ,45

19 We obtained a closed-form formula which prices different options written on the notional index; The notional index is the rate of return on the contributions and is stochastic; This price doesn t depend on the initial capital! This tool allows to put a price on promises to the participants for a better risk management. Issue: How can we finance this guarantees? Who should pay? Contributors? The State?

20 References Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999): Coherent measures of risk. Mathematical finance 9(3), Brigo, D., & Mercurio, F. (2007): Interest rate models-theory and practice: with smile, inflation and credit. Springer. Cochrane, J. H., & Saa-Requejo, J. (2000): Beyond Arbitrage: Good-Deal Asset Price Bounds in Incomplete Markets. Journal of Political Economy 108(1), Cvitanić, J., Pham, H., & Touzi, N. (1999): A closed-form solution to the problem of super-replication under transaction costs. Finance and stochastics 3(1), Frittelli, M. (2000): The minimal entropy martingale measure and the valuation problem in incomplete markets. Mathematical finance: 10(1),

21 References (C td) Henderson, V. (2002). Valuation of claims on nontraded assets using utility maximization. Mathematical Finance, 12(4), Hodges, S. D., & Neuberger, A. (1989). Optimal replication of contingent claims under transaction costs. Review of futures markets, 8(2), Korn, R., & Kraft, H. (2002). A stochastic control approach to portfolio problems with stochastic interest rates. SIAM Journal on Control and Optimization, 40(4), Palmer, E. (2006). What is NDC?, in: R. Holzmann and E. Palmer, eds., Pension Reform: Issues and Prospects for Notional Defined Contribution (NDC) Schemes, (Washington, D.C.: The World Bank), chapter 2. ISBN Pennacchi, G. G. (1999). The value of guarantees on pension fund returns. Journal of Risk and Insurance,

22 Thank you for your attention!